38
Quantum Frequency Combs

Claude Fabre and Nicolas Treps

Sorbonne Université, CNRS, ENS‐Université PSL, Collège de France 4 place Jussieu, 75252 Paris 05, France

This chapter describes the main results that have been recently obtained at the Laboratoire Kastler Brossel at the University Pierre Marie Curie in Paris concerning the quantum properties of optical frequency combs generated by parametric down‐conversion and their use for quantum information processing.

The main interest of this physical system is its ability to generate highly multimode quantum light in the form of trains of ultrafast pulses, or “optical frequency combs,” in which the different teeth of the comb are highly entangled. In particular, this system provides a good test‐bench to study Measurement‐Based Quantum Computing (MBQC) with continuous variables on large cluster states.

Before going into more details, we would like to thank all the PhD students, post‐docs, and visitors that have very efficiently contributed to the advancement of this line of research in the Kastler Brossel Laboratory in Paris throughout the last decade:

Francesco Arzani, Valentin Averchenko, Benoit Chalopin, German De Valcarcel, Adrien Dufour, Giulia Ferrini, Yuri Golubev, Tania Golubeva, Clement Jacquard, Jinxia Feng, Renne Medeiros de Araujo, Giuseppe Patera, Olivier Pinel, Pu Jian, Jonathan Roslund, Roman Schmeissner, Shifeng Jiang, Valerian Thiel, Yin Cai, and Young‐Sik Ra.

38.1 Introduction

It is now well established that nonclassical quantum states of light are valuable candidates for the development of quantum information processing protocols. They have indeed many advantages: they have weak environmental disturbances and can be easily manipulated and detected with low losses. Most noticeably they can often be generated in a nonconditional way and are readily scalable to large numbers of degrees of freedom. For these reasons, many proof‐of‐principle experiments have been demonstrated that utilize either optical qubits in the discrete variable (DV) regime (1) or fluctuations of the quantized electric field in the continuous variable (CV) regime (2). In order to implement a universal set of quantum logical operations, an interaction among the various photonic channels must be established. While strong nonlinear interactions at the single‐photon level are difficult to achieve, it is possible to mimic an interaction among photonic channels through the act of measurement. Such measurement‐induced nonlinearities are the basis of linear optical quantum computing (3,4).

An alternative approach, called MBQC, has recently emerged that exploits the act of projective measurement itself as a means for achieving quantum gates (5). More precisely, a quantum logical operation can be realized using a multipartite entangled state (the “cluster state”) consisting of many entangled “nodes,” and by measuring an appropriate physical quantity on the successive nodes. Owing to the multipartite nature of the entanglement, the result of measurements propagates throughout the cluster to the other nodes in a deterministic manner in a way that implements the desired logical gate.

Optical cluster states, in which the different nodes consist of different modes of the optical electromagnetic field, have been successfully constructed both in the DV (1) and CV ( 2,6) regimes. CV entanglement is of particular interest because the unconditional nature of the quantum state generation allows for both high signal‐to‐noise ratios and high data transfer rates. The traditional methodology to construct CV clusters is to introduce a series of independent squeezed states of light into a linear optical network that is arranged in such a way as to produce the desired entanglement (7). Each node contained within these states, however, necessitates its own source of nonclassical states. Consequently, the incorporation of a large number of such modes rapidly encounters a complexity ceiling in terms of scalability and flexibility.

Alternatively, a multimode source may be exploited in which all of the requisite modes are copropagating within a single beam. The modes can be separated either by some mode‐dispersive technique or through a mode selective measurement, thus avoiding the difficulty of building complex and reconfigurable optical networks. Spatially multimode beams have proven useful for the generation of cluster states when detected with a spatially resolved, multi‐pixel apparatus (8). Another avenue exploits temporal encoding (9): ultrashort light pulses are the nodes of such clusters, and entanglement can be created between a great number of successive light pulses. An alternative promising approach, which is the one that we have chosen to investigate and will describe in this chapter, is to generate entanglement between a great number of frequency modes.

More precisely, we consider “optical frequency combs,” which consist of many monochromatic modes the frequency of which are equally spaced, hence the name. In the time domain, they are periodic trains of identical short pulses of light. They can be generated for example not only by mode‐locked lasers but also by Synchronously Pumped Optical Parametric Oscillators (SPOPOs), and they have been very successfully used for metrological purposes. For such sources, scalability is not a problem: for instance, a commercial mode‐locked laser delivering 100 fs pulses at a rate of 100 MHz is made of a coherent superposition of images frequency modes! In order to generate nonclassical states and entanglement between the frequency modes in such frequency‐combs, a nonlinear process is necessary: optical Kerr effect can been used for such a purpose (10), as well as parametric down‐conversion (11–14).

38.2 Parametric Down Conversion of a Frequency Comb

Let us consider a images nonlinear crystal pumped by a frequency comb of mean frequency images and spacing images , inserted in a resonant cavity of free spectral range also equal to images : this means that the cavity round trip time equals the time interval between successive pump pulses, which then coherently overlap on the nonlinear crystal, hence the name of synchronous pump. Efficient parametric down conversion takes place, in a new configuration where the pump is not monochromatic but consists of a broadband frequency comb (15–19).

Illustration of Parametric downconversion of a femtosecond comb.

Figure 38.1 Parametric downconversion of a femtosecond comb.

In the case of a monochromatic pump of frequency images , the parametric nonlinearity generates a set of EPR‐entangled two‐mode quantum states of frequencies images and images fulfilling images , symmetrically placed around the mid‐frequency images . In the case of a synchronous frequency comb pump, the situation, depicted in Figure 38.1, is more complex: the splitting of a single pump photon of frequency images (pathway 1) creates entanglement between the frequencies images and images , and simultaneously a pump photon of different frequency (pathway 2) correlates frequencies images and images . Quantum correlation is thus established between frequencies images and images by virtue of their mutual link to images . In this manner, every frequency images of the downconverted comb becomes correlated with any other frequency images , among the images “teeth” of the comb.

The Hamiltonian corresponding to a single pass in the crystal that describes the parametric coupling between different cavity modes writes:

38.1 equation

where images , proportional to the square of the pump power, regulates the overall interaction strength and images is the photon creation operator associated with the mode of frequency images . The coupling strength between modes at frequencies images and images is governed by the matrix images , where images is the phase‐matching function and images is the normalized pump spectral amplitude at frequency images .

An alternative description of the state is obtained upon diagonalizing the coupling matrix images . Let us call images the eigenvalues and images the column vector, of components images , giving the decomposition of the corresponding eigenmode (that we will call “supermode”) on the frequency mode basis (20). images is the corresponding annihilation operator, equal to images . The total Hamiltonian is then written in the new basis as a sum of single‐mode squeezing Hamiltonians independently acting on each supermode:

38.2 equation

The eigenspectrum images specifies the degree of squeezing in the uncorrelated squeezed‐vacuum states that constitute the multimode state generated at the output of the nonlinear crystal. Thus, the quantum comb may be described as either an entangled state in the basis of individual frequencies or a set of uncorrelated squeezed states in the supermode basis.

As the synchronous cavity does not spectrally filter the optical state, each supermode is resonant within the cavity, and a standard Optical Parametric Oscillators (OPO) calculation can be used for determining the squeezing for each supermode at the output of the cavity. At the cavity threshold and zero Fourier frequency, the noise of the squeezed quadrature normalized to vacuum is given by:

38.3 equation

images being the eigenvalue of largest modulus of images . There is therefore perfect squeezing in the mode of eigenvalue images , partial quadrature noise suppression for the other modes of images , and unchanged vacuum noise for the modes of null eigenvalues.

Let us stress that the multimode state produced in such a way has a zero mean field and fluctuations around the mean that follow Gaussian statistics: it can be described by a Gaussian Wigner function, which is completely characterized by the covariance matrix images of quadrature fluctuations, that we will consider in more detail in Section (38.4). This multimode Wigner function is everywhere positive.

Figure 38.2 gives the values of the eigenvalues calculated with values of the parameters corresponding to the experiment. It shows that more than 30 supermodes have close eigenvalues, which will give rise to 30 highly squeezed states when the OPO is operated close to threshold.

Illustration of Quadrature noise in the different supermodes calculated with the experimental parameters.

Figure 38.2 Quadrature noise in the different supermodes (normalized to the vacuum noise and in dB scale) calculated with the experimental parameters.

38.3 Experiment

38.3.1 The SPOPO

Illustration of an Experimental setup.

Figure 38.3 Experimental setup.

The experimental setup, described in (21–23), is sketched in Figure 38.3: the laser source is a titanium‐sapphire mode‐locked oscillator delivering images pulses (images FWHM (Full width at half‐maximum)) centered at 795nm with a repetition rate of 76 MHz. This source is frequency doubled in a 0.2 mm bismuth borate (BIBO) crystal (single pass), and the resultant second harmonic pumps an OPO, which consists of a images BIBO crystal contained within a images ring cavity exhibiting a finesse of images . The OPO‐crystal thickness was chosen so that the spectral width of the local oscillator (LO) matches the one of the first supermode. The length of the cavity is locked to the inter‐pulse spacing by injecting a phase‐modulated near‐infrared beam in a direction counter‐propagating to the pump and seed. This locking beam is phase‐modulated at 1.7 MHz with an electro‐optic modulator (EOM), and locking of the cavity length is accomplished with a Pound‐Drever‐Hall strategy. The cavity is operated below‐threshold and in an unseeded configuration. It generates therefore a very weak light beam, with an energy content of a few photons.

Light detection is performed with the help of two possible techniques:

  • a “usual” homodyne detection scheme, shown on Figure 38.3, using silicon photodiodes of high detection efficiency (images detection efficiency, 100 MHz detection bandwidth), with a homodyne visibility of images . The noise level of sidebands situated 1 MHz from the optical carrier is then measured. The cumulative loss of the system is images . Let us stress that such a measurement does not access the quantum noise of each individual pulse, but rather the noise averaged over many successive pulses.
  • a so‐called “multiplexed homodyne” detection scheme: instead of measuring the total intensity of the two output beams of the beamsplitter of the homodyne detector, like in the previous scheme, we have frequency dispersed these two beams with two gratings, measured on two arrays of photodetectors the two spectrally resolved outputs of the beamsplitter and recorded in real time in a computer memory the fluctuations of the difference between the photocurrents of the photodiodes illuminated by the same frequency components (see Figure 38.4).
    Illustration of a frequency-multiplexed homodyne measurement setup.
    Figure 38.4 Sketch of a frequency‐multiplexed homodyne measurement setup.

38.3.2 Experimental Determination of the Full Covariance Matrix

As stated above, the quantum properties of the Gaussian state generated by the SPOPO are fully characterized by its quadrature covariance matrix images , of matrix elements images , where images is either the quadrature images or the quadrature images of the images th mode. It contains therefore all the quadrature correlations images , images , and images .

We are using two different techniques to determine the covariance matrix images of the multimode state under study:

  • The noise properties of the output beam are investigated by “usual” homodyne detection in which the LO pulse form is manipulated with ultrafast pulse shaping methodologies. A 4f‐configuration shaper is constructed in a reflective geometry with a programmable 512 images 512‐element liquid‐crystal modulator in the Fourier plane, which allows us to independently control the amplitude and phase of the diffracted spectrum (24). By varying the relative phase between the shaped LO and the SPOPO output, a measurement is obtained of the images ‐ and images ‐quadrature noise variances images and images for the multimode quantum state projected onto the spectral form of the LO mode images , that is, the diagonal part of the covariance matrix. In order to access the quadrature correlations images for images , one uses as a local oscillator of the homodyne detection the sum of the two modes images and images and thus measures images . The desired matrix element is then calculated using relation:
    38.4 equation

    In the experiment we have used as the experimental mode basis the set of images nonoverlaping frequency bands (images typically ranging from 6 to 10) covering the pump laser spectrum and made successive homodyne measurements with the images frequency bandmodes to measure the diagonal of the covariance matrix, and with the images couples of modes to deduce its off‐diagonal elements. It has been observed that cross correlations of the form images are absent, which permits the covariance matrix to be cast in a block diagonal form: one block for the images ‐quadrature and one block for the images ‐quadrature.

  • As the previous technique requires to perform a great number of different homodyne measurements, we have also used the “multiplexed homodyne detection” setup, which enables us to store in parallel the instantaneous temporal fluctuations of a given quadrature images in these different frequency bands, from which one directly deduces by a temporal average the second moments images and images .

    This technique allows us to measure in real‐time images second moments, that is, only a part of the images covariance matrix, for example, all the images . A measurement with a images phase change LO yields the images moments of the conjugate variables images . The cross correlations images can be obtained with the help a phase change of the images th frequency band of the LO made by the pulse shaper.

38.4 Experimental Results

Figure 38.5 gives an example of covariance matrix deduced from the homodyne measurements obtained by the first technique, which contains the full information on the quantum properties of the generated Gaussian state within a finite frequency resolution depending on the number of frequency‐band modes used in the experiment (here: 6). It reduces actually to two matrices, one for each quadrature, as the images correlations turn out to be zero. Each matrix reveals significant correlations among the frequency bands of the comb. Let us note that the purity of the state, given by the inverse of the determinant of the covariance matrix, is not unity: owing to losses in the OPO, the state under consideration is indeed a statistical mixture.

Illustration of experimental result: difference between the experimental covariance matrix and the covariance matrix of vacuum.

Figure 38.5 Example of experimental result: difference between the experimental covariance matrix and the covariance matrix of vacuum (actually the identity), separately for the images and images quadratures.

38.4.1 Bipartite Entanglement

Entanglement among various frequency bands is quantitatively assessed with the positive partial transpose (PPT) criterion for continuous variables (25), which probes the inseparability of a given state bipartition. A bipartition is created by dividing the images frequency bands used in the detection into two sets. Given eight distinct spectral bands, 127 different frequency band bipartitions exist, and each of these possible bipartitions is subjected to the PPT entanglement witness, which turns out to be negative for all: every possible bipartition of the multimode SPOPO output is entangled. The absence of any partially separable form implies that the SPOPO output constitutes a highly entangled eight‐partite state in which each resolvable frequency element is entangled with every other component (26). As expected, the downconversion of a femtosecond frequency comb indeed creates a quantum object exhibiting wavelength entanglement that extends throughout the entirety of its structure.

However, the multimode character of the comb and its degree of nonclassicality cannot be simply inferred from such a high degree of bipartite entanglement. It is well known that the bipartition of a single‐mode squeezed field creates two entangled modes that satisfy the PPT criteria. We have considered also the same 127 spectral partitions, but now for a simulated single mode field with quadrature values that correspond to those of the first comb supermode. We have found that all of these bipartitions also satisfy the inseparability criterion. Nonetheless, PPT values for the single mode case are weaker than those observed for the comb, which provides a first indication of the comb's multimode character.

38.4.2 Multi‐Partite Entanglement

The output beam of our OPO can also be divided in more than two parties, the maximum being the number images of frequency band modes used in the homodyne detection. For example, for images , there are precisely images different ways to cut the beam into images different parties! For a precise characterization of the quantum state under study, it is important to investigate whether it is entangled or separable with respect to these different multipartitions. For this purpose, we have used the entanglement tests developed by the Rostock team (27) for multimode Gaussian states, in terms of an optimized linear combination of matrix elements of the covariance matrix. These tests have been applied to a 10‐mode experimental covariance matrices (28). They have shown that all the possible multipartitions are fully entangled, more precisely that the density matrix of the state under study is not a convex superposition of product of K density matrices defined in the subspaces constituting the partition with images . This means that the SPOPO beam is a possible tool for multi‐partite quantum information processing involving many parties.

This analysis can be pushed even further (29), as the complexity of the separability problem increases substantially when one studies multipartite systems: one has a rapidly increasing number of choices in the bunching of parties on which one searches for a possible factorization. Hence, a separable state may exhibit a much richer and complex structure of possible convex combinations between the different parties (30–32).

A state that is not a statistical mixture of bipartite factorized density matrices is called “genuinely” multipartite entangled, and genuine entanglement implies multipartite entanglement for every other separation of the modes. However, if a state does not exhibit this specific kind of entanglement, no conclusions on other forms of multipartite quantum correlations can be drawn. In particular, when the number images of parties is more than 2, the density matrix of a multimode state can be a convex combination not only of product states involving images parties but also of combinations of multimodal partitions involving different numbers of parties.

The optimized entanglement criteria mentioned in the previous paragraph can be extended to these more complex combinations. We have used them to characterize a six‐frequency band experimental quantum state, which features 31 bipartitions, 90 tripartitions, 65 four‐partitions, 15 five‐partitions, and 1 six‐partition. The analysis shows that no detectable bipartite entanglement exists in this specific quantum frequency comb, but also that the state is entangled for any higher number of parties. Therefore, the SPOPO state is entangled with respect to any individual bipartition, even though it cannot be identified as a bipartite entangled state: this subtle structure of multipartite entanglement is invisible for genuine entanglement probes.

38.4.3 Extraction of Principal Modes

Frequency band modes constitute a set of orthonormal modes provided there is no overlap between them. They are indeed a basis on which a given solution of Maxwell equations can be decomposed in the case where its spectrum varies slowly compared to the bandwidth of the frequency band modes (around 1 nm in wavelength units in our case). An important property of multimode quantum states is that they have different expressions according to the different mode bases used in their description. For example, it is well known that a two‐mode quantum state appearing as a factorized two‐mode squeezed state on a mode basis images (one squeezed on the images quadrature, the other on the images quadrature) appears also as an entangled Einstein Podolsky Rosen (EPR) state on the mode basis images , where images . Therefore, when the choice of the mode basis for the multimode state is left totally free, meaning that there is no natural “Alice/Bob” bipartition in the considered quantum state, entanglement and factorization of nonclassical states are just two sides of the same coin, which witness the existence of a single quantum resource. This means that in the present case, it is important to look for other mode bases than the frequency band one, in order to find modes which allow to simplify the description of the quantum state. Such modes, which are more “physical” than the initial ones, are often called “principal modes.”

As a covariance matrix is a real symmetric one, it is tempting to diagonalize it. However, the transformation on the quadratures leading to the diagonal form is not a mode basis change, so that it is not possible to find by this way the modes in which the quantum state is described as a product of uncorrelated squeezed modes. More precisely, in our specific case of decorrelated images and images covariance matrices, it is observed that these matrices do not commute. Although the individual images and images block eigenvectors are quite similar, they are not exactly equal. This implies that a common mode basis is not able to simultaneously diagonalize the two quadrature blocks.

To extract relevant modes, one must consider the complete decomposition of the symplectic matrix responsible for the generation of the multimode state. The Bloch‐Messiah reduction (33,34) allows any symplectic transformation to be decomposed into an initial basis change, a perfect multimode squeezer, and a final basis change. When the input state of this transformation is vacuum, the first basis rotation is useless, and the resultant multimode state is pure and may be understood as an assembly of factorized squeezed vacua in a given eigenbasis. However, when the input state either contains classical noise or is not pure, both of these basis rotations become meaningful. Application of the Bloch‐Messiah reduction to a covariance matrix reveals the Williamson (or “symplectic”) eigenvalues as well as the mode structures both for the uncorrelated classical noise sources and for the independent quantum squeezers. These Williamson eigenvalues indicate the existence of residual classical noise on the input state. Importantly, in the presence of excess classical noise, the quantum squeezer basis and the supermode basis do not necessarily correspond. In the present experiment, the input state to the cavity is vacuum, which implies that residual classical noise is introduced by loss mechanisms. Correspondingly, the fact that the images and images blocks of the covariance matrix are not diagonalized by a common basis indicates that the loss mechanism is spectrally dependent (e.g., nonuniform transmission profile of the SPOPO output coupler).

A Bloch‐Messiah reduction of the eight‐mode covariance matrix was implemented in order to reveal the full structure of the comb state in a particular experimental case. The Williamson eigenvalues are found to be respectively 2.0, 2.6, 1.5, 1.0, 0.4, 0.3, 0.2, and 0 in dB scale. The purity images is 0.14. In addition, the bases of the classical noise eigenmodes and the squeezed modes are independently uncovered. The squeezed mode basis remains largely unchanged from run to run, whereas the basis associated with the classical noise exhibits a large degree of variation that depends on specific experimental conditions. This effect arises because the classical noise is relatively small compared to the quantum properties of the comb, and the eigenvalues are nearly degenerate. Consequently, the extraction of well‐defined supermodes from the experimental covariance matrix is feasible even though the matrix cannot be placed in a perfectly diagonal form owing to the influence of classical noise.

In practice, the experimental principal modes are recovered with a more pragmatic strategy, made possible by the lack of images correlations. The diagonalization of the block‐diagonal covariance matrix gives us “x‐eigenmodes” and “p‐eigenmodes” of similar shapes exhibiting alternating squeezing and anti‐squeezing. In order to determine the covariance matrix in the most decoupled set of modes, the eight anti‐squeezed eigenmodes are orthogonalized with a Gram‐Schmidt procedure, and the covariance matrix is re‐expressed in terms of this newly orthogonal basis. The resulting matrix is nearly diagonal and contains the squeezing value for each orthogonalized mode on its diagonal.

The mean squeezing spectrum is shown in Figure 38.6 for the situations of 4, 6, and 8 discrete spectral regions. For the leading modes, a larger overall squeezing level is observed for a smaller number of frequency bands. An increase in the number of available bands is needed to replicate the spectral complexity of higher‐order supermodes. In the case of eight frequency bands, up to five squeezed modes are contained within the conglomerate comb structure. The quadrature in which each of these modes exhibits noise reduction (images or images ) alternates between successive modes in agreement with theoretical predictions (15). The conclusion is that, as expected from theory, the SPOPO behaves as an in situ optical device consisting of an assembly of independent squeezers and phase shifters, with a maximum observed squeezing of 6 dB.

Illustration of Mean noise levels and uncertainties (dB) for each of the orthogonalized Gram-Schmidt modes.

Figure 38.6 Mean noise levels and uncertainties (dB) for each of the orthogonalized Gram‐Schmidt modes. The mean eigenspectra are shown for 8 , 6, and 4 frequency bands. The simulated eigenvalues corresponding to 8 frequency bands are shown for comparison (dots without error bars).

Illustration of Amplitude spectra of the successive principal modes.

Figure 38.7 Amplitude spectra of the successive principal modes.

The orthogonalized principal modes that originate from the covariance matrices are shown in Figure 38.7 . The spectral makeup of each retrieved experimental mode has a shape that roughly approximates the form of a Hermite‐Gauss polynomial, which is the predicted supermode profile. However, as mentioned earlier, it becomes evident that the spectral complexity of higher‐order supermodes is only reproducible with an increase in the number of frequency bands. In addition, as the spectral width images of supermodes following a Hermite‐Gauss progression increases with the mode index images as images , the eigenmodes bandwidth rapidly exceeds the local oscillator bandwidth, limited by the bandwidth of the pump laser, resulting in a decrease of the squeezing observed in these modes. Consequently, the current observation of roughly seven squeezed modes does not represent an inherent upper limit to the quantum dimensionality of comb states. Using broader bandwidth LO pulses, increased spectral resolution, and large cavity bandwidths (all achievable experimentally), states possessing as many as images squeezed modes are expected from theory with the actual experimental values of the different parameters.

In order to corroborate the physical interest of the extracted principal modes, each of these modes has been written directly onto the pulse shaper and then used as a local oscillator in the homodyne detection: one finds that each of these modes indeed exhibits squeezing at a level in accordance with that retrieved from the covariance matrix and deserves the name of principal modes for the analysis of the noise content of the SPOPO output.

Let us mention that we have applied the same experimental technique of multiplexed homodyne detection for determining the full noise covariance matrix of a different light source, namely the mode‐locked laser itself used as a pump in the previous experiment (35,36). Its quadrature noise turns out to be well above the standard quantum noise limit, especially for the phase noise. One can then extract from this matrix principal noise modes that allow to precisely characterize the modal noise structure of the pulsed laser.

38.5 Application to Quantum Information Processing

Cluster states are the highly entangled quantum resources needed for MBQC [38], where quantum operations are effected via measurement processes and feed‐forward operations, and can be implemented in both the continuous variable and discrete variable regimes. The discrete variable regime directly deals with quantum states of negative Wigner functions that cannot be simulated by classical computers, but which are generated in a conditional way with low success rates, making it difficult to scale up to many qubits ( 35,40) 95. The continuous variable regime deals with large networks of deterministically generated squeezed and entangled states (29). However, these quantum states have positive Wigner functions and are not sufficient alone for universal quantum computing, which requires an additional non‐Gaussian operation.

38.5.1 Extraction and Characterization of Cluster States

Cluster states consist of a given topology of nodes (Figure 38.8), which actually constitute in the CV case ( 7,37,38) a specific network of electromagnetic field modes. They can be produced by applying a well‐defined modal unitary transformation images on an initial factorized quantum state consisting of squeezed state in each mode. The usual way to experimentally produce cluster states is therefore to independently generate single‐mode squeezed vacua and to mix them in an appropriate linear network involving beamsplitters and phase shifters that implements the desired unitary transformation images ( 6,39, 40).

Our approach is different and directly uses the highly multimode state described and characterized in the previous sections that is present in a single spatial mode (41–43). Mode‐selective homodyne detection is indeed able to directly address the cluster states that are “embedded” in such a beam, without having to physically extract the cluster state from the multimode entangled beam, in a way that implements MBQC protocols. The advantage is that the reconfiguration of the quantum network is easy and fast, as it requires a simple electronics change of the pulse shaper settings and does not need to modify the experimental setup.

More precisely, as the unitary transformation images mathematically corresponds to a general mode basis change, it is possible to reveal the optical network by measuring the multimode beam in the appropriate mode basis. As shown earlier, the parametric down conversion process generates multipartite entanglement in the frequency band basis images (images ). To simplify notations, we write images the column vector of components images . The column vector made of the squeezed experimental principal modes described in the previous sections images can be obtained from the frequency band basis by the modal unitary transformation images , so that images . After applying the unitary transformation images implementing the cluster state network, the set of modes that constitute the network nodes form a column vector images

38.5 equation

Consequently, with every cluster state is associated a unitary matrix images that allows its nodes to be related to the frequency band mode basis. This transformation can be implemented, and the different cluster nodes interrogated, by homodyne measurements with appropriately mode‐shaped local oscillators.

If one wants to achieve error‐free quantum information processing using these clusters, it is essential to have perfectly correlated nodes. The quality of the different internodal correlations is evaluated using a set of nullifiers, defined by the column vector images given by

38.6 equation

Ideally, for error‐free calculation, the fluctuations of each element of the column vector images should be zero ( 38, 7). In 38.6, images and images are, respectively, column vectors containing the amplitude and phase quadratures of the cluster nodes, and images is the adjacency matrix of the graph, which characterizes the connectivity of the cluster state (Figure 38.8). The nullifiers defined by Eq. 38.6 are linear combinations of quadratures of the squeezed modes that can be written in a matrix form, defining the unitary transformation images . These “nullifier modes” are thus obtained by the action of three successive unitary transformations on the frequency band modes: images , images , and images . They are finally particular spectral modes that can be also reached by mode‐shaped homodyne detection.

Illustration of Different four-mode cluster states, and the corresponding adjacency matrices V.

Figure 38.8 Different four‐mode cluster states, and the corresponding adjacency matrices V.

This scheme was exploited to fabricate and characterize different cluster states with nodes that range in number from four to twelve (Figure 38.9). The variances of the corresponding nullifier modes were measured by homodyne detection with a suitable programming of the pulse shaper. The measured variances of the nullifiers where all between 2 and 3 dB below the shot noise limit. This shows that it is possible to extract from the multimode beam a great number of quantum‐correlated cluster states, but with a level of noise reduction that must be largely improved in order to meet the requirements for successful error‐free quantum computation.

Illustration of Nullifier squeezing values of various cluster states in dB scale.

Figure 38.9 Nullifier squeezing values of various cluster states in dB scale. The variances of the images ‐mode linear cluster state (left rectangles) and diagonal square cluster state (right rectangles) are both given for different images values. The cluster states themselves are sketched respectively at the top left and bottom right of the figure The black points are the individual nullifier variances, the rectangles depict the first and third quartiles of the data, the thin black line contained in the rectangle is the nullifier mean, and the black whiskers indicate the upper and lower extrema of the nullifier collection. All of the nullifier variances are below the shot noise limit, which implies successful generation of the targeted cluster states.

38.5.2 Simulating a Multipartite Quantum Secret Sharing

Quantum secret sharing consists of sharing information (either quantum or classical) between several players using entangled quantum states. The information is first transferred to a multipartite entangled state. Each player is then given a piece of the total entangled state, and the original information can only be retrieved through a collaboration of subsets of the players. The quantum correlations increase both the protocol security and its retrieval fidelity as compared to what is attainable with only classical resources [36 38].

We have simulated on our scheme a five‐partite secret sharing protocol that uses a six‐mode all‐optical quantum network (39) with a six‐node graph structure shaped like a pentagon plus a central node connected to all others (44). The nodes on the edge of the pentagon (labeled 1 to 5) represent the players, and the central node (6), called the dealer, encodes the secret prior to its coupling to the conglomerate state. The nodes corresponding to the players and the dealer are associated with modes that, in turn, are constructed as a combination of the leading six squeezed eigenmodes.

Given this configuration, at least three players must collaborate to reconstruct the secret. Any set of three players constitutes what is termed an access party. As an example, we consider the access party of players 1, 2, and 3. In order to access and therefore reconstruct the quadrature of the secret state, the three players within this access party must each measure a specific quadrature of their local variables and combine their independently obtained results with the dealer's p quadrature measurement in the following access party operators:

38.7 equation
38.8 equation

Here we denoted with the suffix images the observables associated with the network nodes. images , and images are real‐valued coefficients, the exact values of which can be computed so that the expression of the access parties depends only on the secret mode and on the input squeezing. Doing so, if the squeezing goes to infinity, the combined measurement leads to an exact reconstruction of the secret. Furthermore, it can be shown that with only two players, it is not possible to find coefficients so that the combined measurement only depends on the squeezed quadratures. The contribution of the antisqueezed quadratures can never be canceled, and the better the squeezing is, the lower will be the amount of information on the state that only two players can reconstruct. Using pulse‐shaped local oscillator, we could directly measure the access party operators for each possible combination of three players. We have shown significant improvement on the fidelity of the input state reconstruction compared to what can be done with only coherent light. Furthermore, we could experimentally demonstrate that with better squeezing, we indeed increase the reconstruction fidelity.

38.5.3 Toward Measurement‐Based Quantum Computation

Let us stress again that the creation of cluster graph states with our system does not necessitate any change in the optical architecture. Rather, the connectivity of the network structure is varied by simply modifying the basis in which the state is detected. Furthermore, this approach allows for the implementation of any quadratic Hamiltonian, modulo the available resources that are the squeezing eigenvalues. In that sense, our system consists in a first step toward a quantum simulator as it allows for probing any multimode system with quadratic evolution.

However, the present scheme is not compatible with MBQC, as only one mode can be measured at a time before the quantum state is destroyed. To achieve this goal, the multiplexed homodyne detection described in Section 38.3 can be used in conjunction with post processing. This architecture has been theoretically demonstrated to be a versatile universal Gaussian MBQC system (42, 43). Finally, any quantum computing application demonstrating quantum supremacy requires going beyond the Gaussian statistics, which can be efficiently simulated with a classical computer. In our system, non‐Gaussian operation can be implemented using mode‐selective sum frequency generation (45), which allows for mode‐dependent photon subtraction (46) when a photon is detected in the sum‐frequency mode. The implementation of such a scheme (47) will turn our system into a unique highly versatile multimode non‐Gaussian source compatible with MBQC applications.

38.6 Application to Quantum Metrology

38.6.1 Mean Field and Detection Modes

Nonclassical states of light can also be used to push further the quantum limits to the accuracy of measurements of various physical quantities. In the continuous variable regime, it has been shown that EPR‐entangled and squeezed states can be used to improve the sensitivity of optical measurements with respect to small changes of a parameter images of interest (position, time, frequency …). If one restricts himself or herself to the use of multimode Gaussian quantum states images , the Quantum Cramer‐Rao bound in the optical measurement of parameter images depends actually on the properties of two well‐defined optical modes (48,49):

  • the “mean field mode” images , which is the normalized spatial and temporal shape of the mean value of the electric field operator in the quantum state images :
    38.9 equation
    where images is the mean photon number in the quantum state;
  • the “detection mode” images , which is the normalized derivative with respect to images of the mean field mode:
    38.10 equation

    images being a factor ensuring the normalization to unity of the detection mode.

The analysis of Refs ( 49,50) gives the following value to the Quantum Cramer‐Rao bound images , which is the smallest detectable variation of the parameter images , minimized over all possible optical measurements and all possible data‐processing techniques, in the case where the mean photon number in the Gaussian pure quantum state images is large:

38.11 equation

where images is the mean root square of the quadrature noise of the detection mode in phase with the mean field and normalized to shot noise. This means that the noise in such a measurement is entirely due to the quantum noise of the detection mode, not to the noise of the mean field mode. If one uses a coherent state in the measurement, images : he or she gets the usual shot noise limit. One can reduce further the Cramer‐Rao bound using a vacuum‐squeezed state in the detection mode. The present discussion generalizes the well‐known result obtained by Caves and now used in the gravitational wave interferometer Laser Interferometer Gravitational Observatory (LIGO): a phase measurement involving a Michelson interferometer is improved, not by reducing the noise of the input laser (the mean field mode) but by shining squeezed vacuum through the second input port of the interferometer (the “detection mode”).

In addition, it has been shown that it is always possible to reach the Quantum Cramer‐Rao bound using a homodyne detection with the detection mode as the local oscillator mode.

38.6.2 Quantum Metrology with Quantum Frequency Combs

Frequency combs are well known to be almost ideal tools for the metrology of frequencies. They can also be used as clocks and also in range finding. In all cases, they provide ultra‐accurate measurements of frequency images , time images , or distance images , often at the shot noise limit.

The Quantum Cramer‐Rao limit of these measurements can in principle be reached using the approach developed in the previous paragraph: the mean field mode is simply the train of pulses emitted by the laser, and the detection mode, different of course for images , images , or images , can always be expressed on the frequency band basis, and therefore implemented on the LO mode of the homodyne detection by a given pulse shaping operation. We have indeed performed measurements close to the shot noise limit using mode‐locked laser as the light source and homodyne detection using a mode‐shaped local oscillator in the appropriate detection mode. For example, a range finding accuracy of about images has been measured. More information concerning these measurements can be found in (51,52).

In addition, it turns out that in the three considered measurements, the detection mode is close to the second principal mode contained in the SPOPO beam (top line right in Figure 38.7), which is significantly squeezed in our experimental configuration. This means that if one mixes on a highly transmitting beamsplitter, the mode‐locked laser beam and the output of the SPOPO to produce the multimode light beam that is used in the measurement of parameter images , the noise floor of the measurement will be reduced because one will take advantage of the squeezed quadrature in the detection mode. For example, an improvement in the signal to noise ratio of 20% has been observed in the measurement of distance and frequency.

38.7 Conclusion

Because of the relative facility to modify at will the spectral/temporal shape of femtosecond lasers, we have shown that we can precisely analyze, control at the quantum level, and use light beams consisting of thousands of frequency modes. Though the path leading to the use of such experimental techniques for useful quantum information processing and computing is long, as it requires the improvement of the generated multimode squeezing and the implementation of efficient non‐Gaussian operations, the alley of research that has been described in this chapter offers a promising and novel way for the use of highly multimode quantum light in scalable and practical quantum information processing.

Acknowledgment

Laboratoire Kastler Brossel, of the Ecole Normale Supérieure, Sorbonne Université and Collège de France, is associated with the Centre National de la Recherche Scientifique. This work was supported by the European Research Council starting grant Frecquam and the European Union grant QCUMBER (No. 665148).

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